Quantitative Homogenization and Convergence of Moving Averages

Abstract

We study homogenization it its most basic form $$-\left(a\left(\frac{x}{\varepsilon}\right) u_{\varepsilon}'(x)\right)' = f(x) \quad \mbox{for} ~x \in (0,1),$$ where $a(\cdot )$ is a positive $1-$periodic continuous function, $f$ is smooth and $u_{\epsilon }$ is subjected to Dirichlet boundary conditions. Classically, there is a homogenized equation with $a(\cdot )$ replaced by a constant coefficient $\overline{a}>0$ whose solution $u$ satisfies $\Vert u-u_{\epsilon }{\Vert }_{L^{\infty }}\lesssim \epsilon$. We show that local averages can result in faster convergence: for example, if $a(x)=a(1-x)$, then for $x\in (\epsilon ,1-\epsilon )$ $$\left|\frac{1}{\epsilon }{\int }_{x-\epsilon /2}^{x+\epsilon /2}{u}_{\epsilon }(y)dy-u(x)\right|{\lesssim }_{a,f}{\epsilon }^2.$$ If the condition on $a(\cdot )$ is not satisfied, then subtracting an explicitly given linear function (depending on $a(\cdot ),f$) results in the same bound. We also describe another approach to quantitative homogenization problems and illustrate it on the same example.